Analytical Approximate Solution of Schrödinger Equation in D Dimensions with Quadratic Exponential-Type Potential for Arbitrary l-state

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1 Commun. Theor. Phys. 61 ( Vol. 61, No., April 1, 01 Analytical Approximate Solution of Schrödinger Equation in D Dimensions with Quadratic Exponential-Type Potential for Arbitrary l-state Akpan N. Ikot, 1, Oladunjoye. A. Awoga, 1 Hassan Hassanabadi, and Elham Maghsoodi 1 Theoretical Physics Group, Department of Physics, University of Uyo, Uyo, Nigeria Department of Basic Sciences, Shahrood Branch, Islamic Azad University, Shahrood, Iran (Received October, 013; revised manuscript received January, 01 Abstract We present the bound state solution of Schrödinger equation in D dimensions for quadratic exponential-type potential for arbitrary l-state. We use generalized parametric Nikiforov Uvarov method to obtain the energy levels and the corresponding eigenfunction in closed form. We also compute the energy eigenvalues numerically. PACS numbers: Ge, Ca, w Key words: Schrödinger equation, Nikiforov Uvarov method, bound state ndemikotphysics@gmail.com c 01 Chinese Physical Society and IOP Publishing Ltd 1 Introduction The solution of Schrödinger equation in quantum mechanics is very difficult to solve for some central potentials. [1 However, the analytical solutions are possible only in a few simple cases such as harmonic oscillator, [ and the hydrogen atom. [3 In addition, it is a well-known fact that the exact solution of the Schrödinger equation plays an important role in quantum mechanics since they contain all the necessary information regarding the quantum system under consideration. [ In recent times, the study of exponential-type potential has attracted a lot of interest of different authors. [5 8 These potentials under investigation include the Manning Rosen potential, [9 the Hulthen potential, [10 the Eckart potential [11 the five parameter exponential-type potential, [1 Hylleraas potential, [13 1 and others. [15 For an arbitrary l-state, most quantum systems could only be treated by approximation methods. Nonetheless, many authors have applied different approximation to the centrifugal term and obtained analytical approximation to the l-wave solutions of the Schrödinger equation with exponential-type potentials. [16 3 Various methods have been used to obtain the exact or approximate solutions of the Schrödinger equation for exponential-type potentials. These methods include the supersymmetric (SUSY and shape-invariance method, [ the variational, [5 the standard method, [6 path integral approach, [7 the asymptotic interaction method AIM, [8 the exact quantization rule (EQR, [9 31 the hypervirial perturbation, [3 series method, [33 the shifted 1/N expansion, [3 the algebraic approach, [35 the Nikiforov Uvarov method (NU [36 and others. The NU method had been used in calculating the exact energy values of all bound state for some solvable quantum systems. [37 Motivated by the recent interest in exponential-type potentials, [38 we attempt to study the bound state solution of the Schrödinger equation in D dimensions for a quadratic exponential-type potential. [38 0 V (r = V 0(a e αr + b e αr + c (e αr 1, (1 V 0 is the depth of the potential, α is the range of the potential, a, b, c are adjustable parameters. This potential is one of the best analytical potential models used for the vibrational energy of diatomic molecules. It can also be used in the description of molecular dynamics. [39 0 A form of this potential has been used earlier by Zou et al. [39 Also Eshghi and Hamzavi [0 have used different forms of this potential to obtain the solution of Dirac equation under spin symmetry limit. The purpose of this work is to attempt to study the arbitrary l-state solution of the D-dimensional Schrödinger equation with a quadratic exponential type potential, [0 which has many applications in physics and chemistry. The organization of the paper is as follows: The Nikiforov Uvarov method is presented in Sec.. We present the D-dimensional eigenvalue and the corresponding wave functions in Sec. 3. Results and discussion are given in Sec.. Finally, a brief conclusion is given in Sec. 5. Concept of Nikiforov Uvarov Method The Nikiforov Uvarov method [36 was proposed to solve second-order differential equation of the form ψ (s + τ(s σ(s ψ (s + σ(s σ ψ(s = 0. ( (s

2 58 Communications in Theoretical Physics Vol. 61 With appropriate co-ordinate transformation, s = s(r, σ(s and σ(s are polynomial at most a second order and τ(s is a first degree polynomial. The parametric form of the Schrödinger-like equation can be written for any potential as [1 d ψ ds + α 1 α s dψ s(1 α 3 s ds + 1 s (1 α 3 s [ ξ 1s + ξ s ξ 3 ψ(s = 0. (3 According to NU method, the eigenfunction and the corresponding energy eigenvalues equation becomes ψ(s = s α1 (1 α 3 s α1 α13/α3 P (α10 1,(α11/α3 α10 1 n (1 α 3 s, ( (α α 3 n + α 3 n (n + 1α 5 + (n + 1 ( α9 + α 3 α8 + α7 + α 3 α 8 + α 8 α 9 = 0, (5 α = 1 α 1, α 5 = (α α 3, α 6 = α 5 + ξ 1, α 7 = α α 5 ξ, α 8 = α + ξ 3, α 9 = α 3 α 7 + α 3 α 8 + α 6, α 10 = α 1 + α + α 8, α 11 = α α 5 + ( α 9 + α 3 α8, α 1 = α + α 8, α 13 = α 5 ( α9 + α 3 α8. (6 3 D-Dimensional Schrödinger Equation and Solutions The Schrödinger equation for a spherically symmetric potential in D-dimensional [ reads µ [ D + V (rψ nlm(r, Ω m = E nl ψ nlm (r, Ω m, (7 the laplacian operator is defined as D = 1 [ r D 1 r D 1 Λ D (Ω D r r r, (8 V (r is the potential, µ is the reduced mass, is the reduced Planck constant, E nl is the energy spectrum and Ω D represents the angular coordinate. The hyperspherical harmonic functions are the eigenfunction of the operator Λ D (Ω D. Thus, we write ψ nlm (r, Ω m = R nl (ry m l (Ω D, (9 Y m l (Ω D are the hyperspherical harmonic and R nl (r is the hyper radaial wave function. It is well-known that Λ D (Ω D/r is a generalization of the centrifugal barrier for the D-dimensional space and involves the angular co-ordinate (Ω D and the eigenvalues of the hyperspherical harmonic functions Λ D (Ω D are given by Λ D(Ω D Y m l (Ω D = l(l + D Y m l (Ω D, (10 l is the arbitrary angular momentum quantum number. By choosing a common ansatz for the wave function in the form R nl (r = r (D 1/ U nl (r (11 reduces Eq. (7 into the Schrödinger equation in D dimensions as [ d U nl (r dr + µ [E V (ru nl(r + 1 [ (D 1(D 3 r + l(l + D U nl (r = 0. (1 To study any quantum physical system, we solve the D-dimensional Schrödinger equation given in Eq. (1. Substituting Eq. (1 into Eq. (1 leads to the following radial Schrödinger equation d U nl (r { µe dr + µv [ 0 a e αr + b e αr + c (e αr ( (D 1(D 3 } r + l(l + D U nl (r = 0. (13 In order to solve Eq. (13 for l 0, we need to apply the approximation scheme to the centrifugal term given by [ 1 r α[ e αr (e αr 1. (1 Substituting Eq. (1 into Eq. (13 and introducing the change in variable through, s = e αr, we obtain the following compact hypergeometric equation d U nl (s 1 du nl ds + s(s 1 ds + 1 s (s 1 [ (ε + βs + (ε γs (ε + ϕu nl (s = 0, (15

3 No. Communications in Theoretical Physics 59 ε = µe α + 1 [ (D 1(D 3 + l(l + D, (16 1 β = µav 0 α, γ = µbv [ 0 (D 1(D 3 α + l(l + D ϕ = µcv 0 α. Now comparing Eq. (15 with Eq. (3, we find the following parameters: (17, (18 α 1 = α = α 3 = 1, ξ 1 = ε + β, ξ = ε γ, ξ 3 = ε + ϕ. (0 Using Eq. (16, we determine the remaining co-efficient as α = 0, α 5 = 1, α 6 = ε + β + 1, α 7 = ε + γ, α 8 = ε + ϕ, α 9 = β + γ + ϕ + 1, α 10 = 1 + ε + ϕ, α 11 = + ( β + γ + ϕ ε + ϕ, α 1 = ε + ϕ, α 13 = 1 ( β + γ + ϕ ε + ϕ. (1 By using Eq. (1, we obtain the explicit form of the energy eigenvalues of the quadratic exponential-like potential as E nl = α {[ ϕ β (n + σ 1 [ (D 1(D 3 } + + l(l + D + cv 0, ( µ (n + σ 1 (19 σ = 1 [ 1 + (β + γ + ϕ + 1. (3 Referring to Ref. [1, we now turn our attention to the calculation of the corresponding wave function. The explicit form of the weight function becomes ρ(s = s µ (1 s ν, ( µ = 1 + ε + ϕ, ν = ( β + γ + ϕ + 1/ and this gives the first part of the wave function in the form of the Jacobi polynomials as χ n (s = P (µ,ν n (1 s. (5 Also, the second part of the wave function can be found as φ(s = s (µ 1/ (1 s (ν+1/. (6 Thus, the unnormalized wave function expressed in terms of the Jacobi polynomials reads U nl = N n s (µ 1/ (1 s (ν+1/ P (µ,ν n (1 s, (7 and hence the total radial parts of the wave function expressed in terms of the hypergeometric function are R nl (r = N nl r (D 1/ (e αr (µ 1/ (1 e αr (1+ν/ F 1 ( n, n + (µ + ν; µ; e αr, (8 N nl is a normalization constant. The Jacobi polynomial and hypergeometric function are related by the relation [3 P (a,b n (1 x = F 1 ( n, n + a + b + 1; a + 1; x, (9 F 1 (ν, µ, γ, x = Γ(γ Γ(νΓ(µ k=0 Γ(ν + kγ(µ + k x k Γ(γ + k k!. (30 Numerical Results and Discussions In order to test the accuracy of our results. We calculate the energy eigenvalue for various n and l for D = 3,, 5 with three different values of the parameters α as shown in Tables 1, we have set µ = = 1 in our calculation. We also plot the potential V (r as a function of r for different values of the parameters α in Fig. 1. The approximation of Eq. (38 and the centrifugal term 1/r is compared in Fig. which provides us with an alternative approximation.

4 60 Communications in Theoretical Physics Vol. 61 However, the energy eigenvalues obtained in Tables 1 for this potential using the approximation are valid for short range potential and are in good agreement with the other authors. We have used the NU method and solved the radial Schrodinger equation for the quadratic exponential-type potential with arbitrary l-state in D dimensions. We have derived explicitly, the energy eigenvalues and the corresponding wave function. We can study the special cases of this potential..1 Woods Saxon Potential Setting a = 0, b = 1, c = 0 in quadratic exponential-like potential, we obtain Woods Saxon potential V (r = V 0 (1 + e αr. (31 Hence, the energy eigenvalues for l 0 and the wave function for the Woods Saxon potential becomes E nl = α {[ (n + σ1 1 [ (D 1(D 3 } + l(l + D, (3 µ 1 R nl (r = N nl r (D 1/ (e αr (µ 1/ (1 e αr (1+ν/, (33 γ 1 = µv [ 0 (D 1(D 3 α + + l(l + D, β 1 = µv 0 α, µ 1 = 1 + ε, σ 1 = 1 [ 1 + (β1 + γ (, ν 1 = β 1 + γ The eigenvalues and the corresponding wave function of the standard Woods Saxon for D = 3 becomes E nl = α {[ (n + σ1 µ µ 1 = 1 + ε and ν 1 = β 1 + γ 1 + 1/.. Hulthen Potential. (3 1 1 l(l + 1 }, (35 σ 1 = 1 [1 + (β 1 + γ 1 + 1, (36 β 1 = µv 0 α, γ 1 = µv 0 + l(l + 1, (37 α R nl (r = N nl r 1 (e αr (µ1 1/ (1 e αr (1+ν1/, (38 The energy spectrum and the wave function are obtained from Eqs. (1, ( and (8 by setting a = 0, b = 1, c = 1, V 0 = V 0 as V (r = V 0 e αr 1 e αr, {[ E nl = α µ ϕ (n + σ + (n + σ (39 1 [ (D 1(D 3 } + l(l + D + V 0, (0 1 R nl (r = N nl r (D 1/ (e αr (µ 1/ (1 e αr (1+ν/, (1.3 Generalized Morse Potential γ = µv [ 0 (D 1(D 3 α + + l(l + D, ( ϕ 1 = µv 0 α, σ 1 = 1 [ 1 + (γ + ϕ + 1, (3 µ = 1 + ( ε + ϕ, ν = γ + ϕ + 1. ( The generalized Morse potential is defined by Dong [, as δ, V (r = D e (1 (5 e αr 1

5 No. Communications in Theoretical Physics 61 δ = e αrc 1 and the three positive parameters D e, r c, and α denote the dissociation energy, the equilibrium inter nuclear distance, and the range of the potential well respectively. We can re-write Eq. (5 in the form, V (r = D e [ e αr (1 + δe αr + (1 + δ (e αr 1. (6 Now comparing Eq. (1 with Eq. (6 gives the following positive parameters: V 0 = D e, a = 1, b = (1 + δ, c = (1 + δ. (7 Substituting these parameters into Eq. ( and Eq. (8, we obtain the eigenvalues and the wave function for the generalized Morse potential in D dimensions as, E nl = α {[ ϕ3 β 3 µ (n + σ 3 + (n + σ 3 1 [ (D 1(D 3 } + l(l + D + (1 + δ D e, (8 1 R nl (r = N nl r (D 1/ (e αr (µ3 1/ (1 e αr (1+ν3, (9 σ 3 = 1 [ 1 + (β3 + γ 3 + ϕ 3 + 1, (50 β 3 = µd e α, (51 γ 3 = µ(1 + δd [ e (D 1(D 3 α + + l(l + D 3, (5 µ 3 = 1 + ε + ϕ 3, (53 ν 3 = This result is consistent with those obtained by Dong [ for D = 3. β 3 + γ 3 + ϕ (5 Fig. 1 A plot of the potential as a function of r for V 0 = 0.5 MeV, a = 0.0, b = 0.0, c = 0.03 with various values of α = 0.05, 0.1, and 0. fm 1. Fig. A plot of the variation of the centrifugal term 1/r and the approximation as a function of r for various values of α = 0.05 and 0.5 fm 1. 5 Conclusions In this paper, we have studied the approximate bound state solutions of the Schrödinger equation in D dimensions for the quadratic exponential-like potential for arbitrary l-state by means of the NU method by exploring the approximate scheme to deal with the centrifugal barrier term. some numerical results are given for the energy eigenvalues in Tables 1 for D = 3,, 5. We have also plotted the potential as a function of r for two different parameters of α in Figs. 1 and shows the variation of the approximation scheme and the centrifugal term, respectively. The special case of this potential Woods-Saxon, Hulthén and the generalized Morse potential are discussed and are in consistent with those in the literature. [ Finally, interested readers may consult Refs. [5 7 for the related work and Ref. [8 for the wave equation in higher dimensions and other works on supersymmetric quantum mechanics (SUSYQM [9 50 for comparison.

6 6 Communications in Theoretical Physics Vol. 61 Table 1 Energy eigenvalues for D = 3,, and 5 with α = 0.01 and a = 0.0, b = 0.0, c = n 1 E nld D = 3 D = D = Table Energy eigenvalues for D = 3,, and 5 with α = 0.03 and a = 0.0, b = 0.0, c = n l E nld D = 3 D = D = References [1 O.A. Awoga and A.N. Ikot, Pramana J. Phys. 79 ( [ L.I. Schliff, Quantum Mechanics, 3 rd ed., McGraw-Hill, New York (1995. [3 L.D. Landau and Lifshitz, Quantum Mechanics-Non- Relativistic Theory, 3 rd ed., Pergamon, New York (1977. [ S.H. Dong, Commun. Theor. Phys. 55 ( [5 W.C. Qiang and S.H. Dong, Phys. Lett. A 37 ( [6 S.M. Ikhdair, Int. J. Mod. Phys. C 0 ( [7 N. Saad, Phys. Scr. 76 (007 6, [8 S.M. Ikhadair and R. Sever, Int. J. Mod. Phys. C 19 ( [9 A.D. Antia, A.N. Ikot, and L.E. Akpabio, Euro. J. Sci. Res. 6 ( [10 A.N. Ikot, L.E. Akpabio, and E.J. Uwah, Electr. J. Theor. Phys. 8 ( [11 W.C. Qiang and S.H. Dong, Phys. Lett. A 368 ( [1 A.N. Ikot, L.E. Akpabio, and J.A. Obu, J. Vect. Relat. 6 (011 1.

7 No. Communications in Theoretical Physics 63 [13 A.N. Ikot, O.A. Awoga, and B.I. Ita, Few-body Syst. Doi: /s y (01. [1 A.N. Ikot, Chin. Phys. Lett. 9 ( [15 A.N. Ikot, O.A. Awoga, and A.D. Antia, Chin. Phys. B ( [16 S.H. Dong, W.C. Qiang, G.H. Sun, and V.B. Bezerra, J. Phys. A 0 ( [17 G.F. Wei, C.Y. Long, and S.H. Dong, Phys. Lett. A 37 ( [18 S. Dong, J. Garcıia-Ravelo, and S.H. Dong, Phys. Scr. 76 ( [19 R.L. Greene and C. Aldrich, Phys. Rev. A 1 ( [0 S.H. Dong and X.Y. Gu, J. Phys.: Conf. Ser. 96 ( [1 G.F. Wei, C.Y. Long, X.Y. Duan, and S.H. Dong, Phys. Scr. 77 ( [ S H. Dong, W.C. Qiang, and J. Garcıia-Ravelo, Int. J. Mod. Phys. A 3 ( [3 C.S. Jia, T. Chen, and L.G. Cui, Phys. Lett. A 373 ( [ B. Gönul and I. Zorba, Phys. Lett. A 69 ( [5 E.D. Ficho and R.M. Ricotta, Phys. Lett. A 69 ( [6 A.N. Ikot, L.E. Akpabio, and A.D. Antia, Arab. J. Sci. Eng. Doi: /S (011. [7 S.M. Ikhadair and R. Sever, Annalen der Physik (Berlin 17(11 ( [8 O. Bayrak and I. Boztosun, J. Phys. A 39 ( [9 Z.Q. Ma and B.W. Xu, Euro. Phys. Lett. 69 ( [30 F.A. Serrano, X.Y. Gu, S.H. Dong, and Q. Dong, J. Math. Phys. 51 ( [31 W.C. Qiang and S.H. Dong, EPL 89 ( [3 J.P. Killingbeck, A. Grojean, and G. Jolicard, J. Chem. Phys. 116 (00 7. [33 J. Yu, S.H. Dong, and G.H. Sun, Phys. Lett. A 3 ( [3 S.M. Ikhdair and R. Sever, Int. J. Mod. Phys. A ( [35 M.R. Setare and E. Karimi, Phys. Scr. 75 ( [36 A.F. Nikiforov and V.B. Uvarov, Special Functions of Mathematical Physics, Birkhauser, Basel (1988. [37 M. Hamzavi, A.A. Rajabi, and H. Hassanabadi, Few- Body Syst. 5 ( [38 A. Arda, R. Sever, and C. Tezcan, Chin. J. Phys. 8 ( [39 X. Zou, L.Z. Yi, and C.S. Jia, Phys. Lett. A 36 ( [0 M. Eshghi and M. Hamzavi, Commun. Theor. Phys. 57 ( [1 C. Tezcan and R. Sever, Int. J. Theor. Phys. 8 ( [ H. Hassanabadi, B.H. Yazarloo, S.S. Zarrinkamar, and H. Rahimov, Commun. Theor. Phys. 57 ( [3 I. S. Gradshetyh and I. M. Rhyzhik, Table of Integrals, Series and Products, Elssevier, Burlington (007. [ Z.H. Deng and Y.P. Fan, Shandong University J. 7 ( [5 G.F. Wei, S.H. Dong, and V. Bezerra, Int. J. Mod. Phys. A ( [6 W.C. Qiang and S.H. Dong, Phys. Scr. 79 ( [7 W.C. Qiang and S.H. Dong, Phys. Lett. A 37 ( [8 S.H. Dong, Wave Equations in Higher Dimensions, Springer, Dordrecht (011. [9 M.R. Setare and Z. Nazari, Acta Phys. Polo. B 0 ( [50 H. Hassanabadi, E. Maghsoodi, S. Zarrinkamar, and H. Rahimov, Mod. Phys. Lett. A 6 (